Finally, we caution that all estimates presented in this section need to be taken with a grain of salt.
The ad hoc identification of the hyperelectric current with either the electric field times a proportionality factor, \(\mathbf {J} = J/E\,\mathbf {E}\) , or the magnetic field times a proportionality factor, \(\mathbf {J} = - J/B\,\mathbf {B}\) , does not stand on firm theoretical ground.
These two approaches merely serve the purpose to make progress by deriving simple and rough estimates for the efficiency of gauge-field production during axion inflation.
In fact, it has recently been pointed out in Ref. [1]} that a self-consistent mean-field approximation automatically gives rise to both electric and magnetic conductivities.
In the two approaches that we just explained, it is assumed that a particular form of the induced current holds even at the level of perturbations, namely, \(\delta \mathbf {J} = \sigma _B\,\delta \mathbf {B}\)  [Eq. (REF )] or \(\delta \mathbf {J} = \sigma _E\,\delta \mathbf {E}\)  [Eq. (REF )].
However, strictly speaking, this is not fully accurate, since the directions of the perturbed electric and magnetic fields are not necessarily (anti) parallel at the onset of the tachyonic growth around \(k/a \sim 2 \xi H\) .
As a consequence, both conductivities, the electric conductivity \(\sigma _E\)  and the magnetic conductivity \(\sigma _B\) , appear in the perturbed equation of motion.
In Ref. [1]}, following the approach in Ref. [3]}, a dynamical and self-consistent equilibrium solution for the strength of the electric and magnetic background fields is constructed that utilizes both \(\sigma _E\)  and \(\sigma _B\) .
Similar to the GEF estimate, this solution for the electric and magnetic fields lies well above the equilibrium estimate, and relatively close to but below the maximal estimate in Ref. [3]} (see Fig. 2 of Ref. [1]}).
This is one step forward to the complete picture.
Still, the analysis is limited to the stationary equilibrium and needs to be extended to cope with the time-dependent dynamics.
